Ultrasound experiment by M. Fink

Transcription

1 Ultrasound experiment by M. Fink cf. A. Tourin, M. Fink, and A. Derode, Multiple scattering of sound, Waves Random Media (), R3-R6. Experimental set-up for a time-reversal experiment through a multiple-scattering medium: (a) first step, the source sends a pulse through the sample, the transmitted wave is recorded by the TRM. (b) second step, the multiply scattered signals have been time-reverted, they are retransmitted by the TRM, and S records the reconstructed pressure field.

2 Experimental observations The source emits a short µs pulse. The TRM records a long scattered signal. Recompression at the source location after propagation of the time-reversed wave.

3 Scattering of an acoustic pulse in random media Acoustic equations for pressure p and speed u: (p, u) tr ( t ε ) p t + κ(z) u z = ρ(z) u t + p z = ρ(z) = + ν( z ε ) κ (z) = + η( z ε ) (p,u) ref ( t ε ) L z IC: left-going pulse incoming from the right homogeneous half-space. (p,u) inc ( t ε ) m(z) = η(z) + ν(z), n(z) = η(z) ν(z) Local velocity: c(z) = p κ(z)/ρ(z). Local impedance: ζ(z) = ρ(z)c(z).

4 Integral representation of the reflected signal Send a left-going pulse f( t ε ): Reflected signal: p ε inc(t,z = ) = f( t ε ) = π p ε ref(t) = π ρ(z) = + ν( z ε ) z (z) = + η( κ ε ) p ε inc (t) p ε ref (t) L z e iωt ε e iωt ε ˆf(ω)R ε ω )()dω T ε ω (z) ˆf(ω)dω R ε ω(z) is the reflection coefficient for a random slab [ L, z]: Rω ε (z) L z drω ε dz = iω ε m( z ε )Rε ω + iω ε n( z iωz ε )e ε (Rω) ε iω ε n( z iωz )e ε, ε with the initial condition at z = L: R ε ω(z = L) =. Energy conservation R ε ω + T ε ω = uniform boundedness of R ε ω.

5 Numerical simulation 6 Reflected Signal Transmitted Signal 4 8 t 6 4 Incoming Pulse Random Medium z

6 Denote p ε inc(t) = f( t ε ). Record p ε ref(t) up to time t. Time reversal in reflection (TR) ρ(z) = + ν( z ε ) κ (z) = + η( z ε ) (p, u) ε inc(t) (p, u) ε ref(t) L z Cut a piece p ε ref,cut(t) = p ε ref(t)g (t), with supp(g ) [,t ]. Time reverse and send back p ε inc(t R)(t) = p ε ref,cut(t t). (p, u) ε inc(t R)(t) ρ(z) = + ν( z ε ) κ (z) = + η( z ε ) (p, u) ε ref(t R)(t) L z

7 The incoming signal f( t ε ) Expression of the refocused pulse p inc (t) = π e iωt ε ˆf(ω)dω propagates into the medium and generates the reflected signal: p ε ref(t) = e iωt ε ε ˆf(ω)R ω ()dω π Record up to time t and cut a piece of the recorded signal (i.e. G (t) = [,t ](t)) p ε ref,cut(t) = p ε ref(t,z = )G (t) Time-reverse and send back into the medium: p ε inc(t R)(t) = p ε ref(t t)g (t t) = e iω(t t) ε ˆp ε ref(ω )Ĝ( ω ω )dω dω πε ε The signal is real-valued: p ε inc(t R)(t) = πε e iω(t t) ε ˆp ε ref (ω )Ĝ( ω ω )dω dω ε

8 The new signal propagates into the same medium and generates a new reflected signal observed at time t + εs: p ε ref(t R)(t + εs) = ˆp ε inc(t π R)(ω)Rω()e ε iωt ε iωs dω Subsituting the expression of ˆp ε inc(t R) into this equation: p ε ref(t R)(t + εs) = e iωs e iω(t t ) (π) ε ˆf(ω )Ĝ( ω ω ) ε ε R ε ω()r ε ω ()dω dω. Change of variables ω = ω εh: p ε ref(t R)(t + εs) = (π) e iωs e iω(t t ) ε Rω()R ε ω εh ε ()dh dω. ˆf(ω εh) Ĝ (h) The autocorrelation function R ε ω ()R ε ω() plays a primary role. Refocusing at t = t.

9 We have obtained:» E R ε ω+ εh ()R ε ω εh () ε W (,ω,τ)e ihτ dτ In the time domain: E ˆp ε ref(t R)(t + εs) ε = = e iωs ˆf(ω) Ĝ (π) (h)» W (,ω, τ)e ihτ dτ dh dω e iωs ˆf(ω)G (τ)w (, ω,τ)dτdω π e iωs ˆf(ω) ˆKT R (ω)dω π = (f( ) K T R ) (s) where ˆK T R (ω) = G (τ)w (,ω,τ)dτ This results only holds true in average (averaging over all possible realizations of the medium)!

10 Frequency correlation of R ε ω Let us consider the forth-order moment at 4 different frequencies of Rω ε» E R ε ω + εh R ε R ε ω εh ω + εh R ε ω εh»» E R ε ω + εh R ε E R ε ω εh ω + εh R ε ω εh if ω ω. ε

11 p ε ref(t R)(t + εs) = e iωs g(ω,h)r ε ω+ εh R ε ω εh dωdh E ˆp ε ref(t R)(t + εs) = ε.. dω dω dh dh e i(ω +ω )s g(ω, h )g(ω, h )» E R ε ω + εh R ε R ε ω εh ω + εh R ε ω εh.. dω dω dh dh e i(ω +ω )s g(ω, h )g(ω, h )»» E R ε ω + εh R ε E R ε ω εh ω + εh R ε ω εh ε E ˆp ε ref(t R)(t + εs) Var `p ε ref(t R)(t + εs) ε = Convergence in L and in probability of p ε ref(t R)(t + εs). Decorrelation in frequency of R ε ω = Self-averaging in time of p ε ref(t R).

12 Convergence of the refocused pulse The refocused signal `p ε ref(t R)(t + converges in probability as εs) s (, ) ε to P ref(t R) (s) = (f( ) K T R ( )) (s) ˆK T R (ω) = G (τ)w (,ω,τ)dτ where W (,ω, τ) is the deterministic density given by the system of transport equations. In particular, if L : W (,ω,τ) = 8γ n ω (8 + γ n ω τ) The refocused pulse has deterministic center and shape. There is statistical stability.

13 Numerical simulation Refocused Signal t 6 4 Re emitted Signal Random Medium z

14 Comparisons simulations - theory signal.5 signal t.4 Profiles of the refocused pulse at z = Comparison with the input pulse f(t) t Comparison with the asymptotic formula K TR f( t)

15 Application to imagery Goal: extract the information about the large-scale properties of the medium (ρ, κ ) ρ(z) = ρ (z) + ν( z ε ) κ(z) = + η( z κ (z) ε This information is contained in W (, ω,τ). The problem is to find a statistically stable estimator of W. Method: ) Compare the input pulse f(t) and the refocused pulse K TR f( t). Extract ( ˆK TR (ω)) ω. ˆK TR (ω) = G (τ)w (, ω,τ)dτ ) Use different truncation functions G to get (W (, ω,τ)) ω,τ. large-scale variations of the medium. Statistically stable method, no local average is needed.

16 Application to communications How to send a message f(t) from E to R in a highly scattering medium? ) R emits a short, broadband pulse pulse f (t). ) E receives and records a noisy signal G(t). 3) E emits [f G( )](t). 4) R receives [K TR f ( ) f](t). R can extract f. (OK in ocean acoustics, difficult in electromagnetics).

17 Time reversal in changing media Medium i (p, u) ε inc(t) Denote p ε inc(t) = f( t ε ). Record p ε ref(t) up to time t. ρ(z) = + ν i ( z ε ) κ (z) = + η i( z ε ) (p, u) ε ref(t) L z Cut a piece p ε ref,cut(t) = p ε ref(t)g (t), with supp(g ) [,t ]. Time reverse and send it back p ε inc(t R)(t) = p ε ref,cut(t t). Medium ii ρ(z) = + ν ii ( z ε ) κ (z) = + η ii( z ε ) (p, u) ε inc(t R)(t) (p, u) ε ref(t R)(t) L z

18 Changing media The density and compressibility fluctuations (ν i,η i ) and (ν ii, η ii ) are identically distributed. Integrated correlation functions: γ n = E[n()n(z)]dz, γ m = E[m()m(z)]dz Degree of correlation δ [, ]: δ m = R E[m i ()m ii (z)]dz R E[m i ()m i (z)]dz, δ n = R E[n i ()n ii (z)]dz R E[n i ()n i (z)]dz δ = complete correlation. Refocused pulse: p ε ref(t R)(t + εs) = (π) where the reflection coefficient R ε,i ω dr ε,i ω dz = i ω ε m i( z ε )Rε,i ω δ = complete decorrelation. e iωs ˆf(ω εh) Ĝ t (h) Rω ε,ii ()R ε,i ω εh ()dh dω satisfies the Ricatti equation: + iω ε n i( z iωz ε )e ε (Rω ε,i ) iω ε n i( z iωz )e ε ε

19 Asymptotics of the refocused pulse Expectation of the autocorrelation function R ε,i ω R ε,ii E» R ε,ii ω+ εh ()R ε,i ω εh () ε ω : v (,ω,τ)e ihτ dτ where v is given by a system of transport equations for (v p ) p N : v p z + p v p τ = 4 δ nγ n ω p (v p+ + v p v p ) δ n γ n ω p v p ( δ m )γ m ω p v p v p (z = L, ω,τ) = δ (τ) (p) Convergence of the expectation of the refocused pulse: E[p ε ref(t R)(t + εs)] ε e iωs ˆf(ω)G (τ)v (,ω, τ)dτdω π But: E[p ε ref(t R)(t + εs) ] ε» π e iωs ˆf(ω)G (τ)v (, ω,τ)dτdω

20 Equivalently v p (z,ω, τ, z) = E[W p (z, ω,τ)] where (W p ) p N is solution of dw p + p W p τ dz = ipωp ( δ m )γ m W p dw z + 4 δ nγ n ω p (W p+ + W p W p ) δ n γ n ω p W p W p (z = L, ω,τ) = δ (τ) (p) W z is a standard Brownian motion. Thus E[p ε ref(t R)(t + εs)] ε π e iωs ˆf(ω)G (τ)e [W (, ω,τ)] dτdω Convergence of the finite-dimensional distributions: E ˆp ε ref(t R)(t + εs ) p...p ε ref(t R)(t + εs k ) k p ε h Q R R E j k W (,ω, τ) ˆf(ω)e pj i iωs j G (τ)dωdτ π Tightness of (p ε ref(t R)(t + εs)) s (, ) in the space of continuous functions. Conclusion: p ε ref(t R)(t + εs) converges in distribution to R R W (,ω,τ) ˆf(ω)e iωs G (τ)dωdτ π

21 Probabilistic representation of the transport equations Consider our familiar jump (Markov) process (N z ) z L with state space N and generator Lφ(N) = 4 γ nω N (φ(n + ) + φ(n ) φ(n)) Therefore (Feynman-Kac) : τ» W (,ω, τ)dτ = E exp i p «γ m ( δ m )ω N L s dw s τ L exp δ «n γ n ω N L sds L (N ) [τ,τ ] N s ds«n L = where E is the expectation w.r.t the distribution of (N z ) z L L

22 Convergence of the refocused pulse The refocused signal `p ε ref(t R)(t + converges in distribution as εs) s (, ) ε to P ref(t R) (s) = (f( ) K T R ( )) (s) ˆK T R (ω) = G (τ)w (,ω,τ)dτ where W (,ω, τ) is the random density given by the system of transport equations driven by the Brownian motion W z. The refocused pulse has random center and shape. There is no statistical stability.

23 signal. signal t t Fixed medium δ m = δ n = Changing medium δ m = δ n =.75 Refocused pulses Comparison of the refocused pulses from the numerical experiments and the expected refocused pulse shape obtained by the asymptotic theory. Only the density is random κ i = κ ii κ = γ m = γ n.

24 Mean refocused shape E[P ref(t R) (s)] = (f( ) E[K T R ]( ))(s) E[ ˆK T R ](ω) = G (τ)e[w (, ω,τ)]dτ For large slab L : E[W (, ω,τ)] = γ ω δ 8 «tanh δ γ ω τ 8» + δ tanh γ ω τ δ 8 «where γ = γ n + ( δ m )γ m δ = δ n γ n γ n + ( δ m )γ m If G (t) = [,t ](t), «δ tanh γ ω t E[ ˆK 8 T R ](ω) = δ p «δ δ + tanh γ ω t 8

25 Pulse stabilization for a particular class If δ m =, then the signal `p ref(t R) (t + converges in probability εs) t (, ) to P ref(t R) (s) as ε where P ref(t R) is the deterministic pulse shape: P ref(t R) (s) = (f( ) K T R ( )) (s) ˆK T R (ω) = G (τ)w (,ω,τ)dτ For large slab L : W (,ω, τ) = δ nγ n ω 8 «tanh δ n γ n ω τ 8» + δ tanh n γ n ω τ δ n 8 «where δ n is the correlation degree. Sufficient condition for δ m = : the local velocity is not changed by the time-perturbation m i m ii. True in particular for Goupillaud medium.

26 Conclusion Statistical stability of the refocused pulse depends on the statistical properties of the reflection coefficient R. Asymptotic framework ε : - Frequency decorrelation of R. - Moments of R satisfy a system of transport equations. - Representation in terms of a canonical jump Markov process on the nonnegative integers. No statistical stability if the medium is changing (except in special cases).